Foundation Pair Kernel Relation Locality Mean Field Weight Supported On Unconstr

The unconstrained relation result

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition relation describes which pairs of sites in that ledger can influence each other. One foundational goal, called L0, is to show that the framework forces this relation to be local: that any two sites which interact must lie within some fixed distance of each other. The declaration meanFieldWeight_supported_on_unconstrained is a step in that investigation, and its result is a negative one.

The theorem shows that if the recognition relation is unconstrained, meaning it simply relates every site to every other site, then a particular weight graph called the mean-field weight graph is supported on it. In plainer terms, if you allow any two sites to be connected, then the mean-field graph, which connects everything to everything, fits inside that relation. This is a proof, not an assumption, and it is machine-checked.

The consequence is that this unconstrained relation cannot force the locality property L0. The framework proves this directly: for any chosen radius, there exists a finite ledger where the unconstrained relation holds, the mean-field weight graph is supported on it, and yet the graph is not local at that radius. This closes a naive route to proving L0, namely the idea that any recognition structure automatically implies locality. The all-true relation is a dead end.

In Recognition Science, the framework models this as follows. The surviving route to L0 is not an unconstrained relation but a metrically bounded one: a relation where connected sites are guaranteed to lie within a fixed index radius. If such a bounded relation supports all nonzero weights, then the framework proves the locality property follows. A hand-selected band geometry, where sites connect only to their immediate neighbors, is shown to inhabit this route, but that is a model choice, not a forced consequence.

The declaration does not claim that the mean-field weight graph is local, nor that an unconstrained relation is the correct physical choice. It does not prove that any recognition structure forces L0. What it establishes is a precise obstruction: the unconstrained relation is too permissive to do the work. The open obligation is now to force a production recognition relation to be bounded from more primitive dynamics, a task the framework names explicitly as remaining open.

/-- The mean-field weight graph is supported on the all-true relation. -/
theorem meanFieldWeight_supported_on_unconstrained (n : ℕ) :
    RelationSupportsWeight (unconstrainedRelation n) (meanFieldWeight n) := by
  intro i j _hne
  exact trivial

/-- **CLOSED NEGATIVE.** An unconstrained recognition relation supports the
screening mean-field graph, so "any RecognitionStructure ⇒ FiniteRange" is false. -/
theorem unconstrainedRelation_does_not_force_finiteRange (radius : ℕ) :
    ∃ n : ℕ,
      (∀ i j : Fin n, unconstrainedRelation n i j) ∧
        RelationSupportsWeight (unconstrainedRelation n) (meanFieldWeight n) ∧
          ¬ FiniteRange (meanFieldWeight n) radius := by
  refine ⟨radius + 2, fun _ _ => trivial,
    meanFieldWeight_supported_on_unconstrained (radius + 2),
    meanFieldWeight_not_finiteRange radius (radius + 2) (le_refl _)⟩

theorem finiteRange_export_v1
    {n radius : ℕ} {R : Fin n → Fin n → Prop} {G : WeightedLedgerGraph n}
    (hB : BoundedRecognitionRelation R radius)
    (hS : RelationSupportsWeight R G) :
    FiniteRange G radius :=
  boundedRecognitionRelation_supports_finiteRange hB hS

/-- MODEL shadow integration: the hand-selected band carrier inhabits
`finiteRange_export_v1`. This does not force the production relation. -/
theorem bandWeight_inhabits_finiteRange_export_v1 (n : ℕ) :
    BoundedRecognitionRelation (bandRelation n) 1 ∧
      RelationSupportsWeight (bandRelation n) (bandWeight n) ∧
        FiniteRange (bandWeight n) 1 :=
  ⟨bandRelation_bounded n,
    bandWeight_supported_on_bandRelation n,
    finiteRange_export_v1 (bandRelation_bounded n)
      (bandWeight_supported_on_bandRelation n)⟩

What this page does not claim

The mean-field weight graph is local at any radius. An unconstrained recognition relation is the correct physical choice. The framework forces the locality property L0 from any recognition structure.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PairKernelRelationLocality.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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